Averaging for split-step scheme

INSTITUTE OF PHYSICS PUBLISHING Nonlinearity 16 (2003) 1359?1366

NONLINEARITY PII: S0951-7715(03)53701-X

Averaging for split-step scheme

Vadim Zharnitsky1

Mathematical Sciences Research, Bell Laboratories-Lucent Technologies, Murray Hill, NJ 07974, USA

Received 19 September 2002, in final form 4 April 2003 Published 9 May 2003 Online at stacks.Non/16/1359

Recommended by Weinan E

Abstract The split-step Fourier method for solving numerically nonlinear Schro?dinger equations (NLS) is considered as NLS with rapidly varying coefficients. This connection is exploited to justify the split-step approximation using an averaging technique. The averaging is done up to the second order and it is explained why (in this context) symmetric split-step produces a higher order scheme. The same approach is applied to dispersion managed NLS to show that anti-symmetric dispersion maps lead to higher order validity of the corresponding averaged equation.

Mathematics Subject Classification: 78A60, 65M12

1. Introduction

The split-step Fourier method is one of the most widely used numerical schemes for solving nonlinear Schro?dinger (NLS) equations. This is especially true for the problems arising in the fields of nonlinear optics and fibre-optics communication. Under some extreme conditions, however, e.g. near collapse in critical cases or for very short pulses (in high bit-rate systems), the split-step scheme becomes difficult to use. Indeed, short pulses require taking into account a larger number of harmonics and using shorter time-steps, so some other numerical schemes (which are usually slower) are used, such as finite difference on a variable grid. In addition, higher order perturbations of NLS such as the self-steepening term may cause the split-step scheme breakdown.

Therefore, it is important to understand the limits of applicability and the optimal use of the split-step scheme. In this paper, we study the errors resulting from using the split-step scheme for the standard NLS. We consider the so-called semi-discretized system, where the space variable is continuous but the time variable is discrete.

For the split-step scheme we prove the validity of this approximation (for the first- and second-order scheme). For the so-called dispersion managed NLS (DMNLS), which is a

1 Present address: PACM, Princeton University, Princeton, NJ 08544, USA.

0951-7715/03/041359+08$30.00 ? 2003 IOP Publishing Ltd and LMS Publishing Ltd Printed in the UK 1359

1360

V Zharnitsky

well-known model in the field of fibre-optics communication, we show that anti-symmetric dispersion maps lead to a higher order error. This is established by using the similarity between the spilt-step scheme for NLS and DMNLS.

The main idea of the split-step algorithms is to make use of the special structure of an evolutionary equation. For example, consider a PDE ut = F1(u) + F2(u), where the righthand side consists of two easily solvable parts. In other words, we assume that one can easily solve ut = Fi(u), i = 1, 2. A well-known example of this type is the linear Schro?dinger equation

ut = iuxx - iV (x)u,

where ut = iuxx is easily solvable in Fourier space and ut = iV (x)u can be solved in the original x-space.

Then, one can naturally try to approximate the solution of the full equation u(t) = G(t)u0, where G(t) is the evolution operator, by considering (G1(t/N ) G2(t/N ))N u0 where N is

large. It turns out that this procedure does work under some conditions with the error of order O(N -1), see [8] for a more detailed description, for modified schemes and many references.

Note that in the above example, this approximation will take the form

eiH t u0 ei(t/N)x2 e-iV (x)t/N N u0,

where H = x2 - V (x). The NLS equation is given by

iut + uxx + |u|2u = 0

with the split-step approximation given by

u(t ) ei(t/N)x2 ei|u|2(t/N) N u0.

We will justify the Fourier split-step scheme for NLS by using the following observation:

the split-step approximation of the standard NLS is equivalent to the NLS with rapidly varying

coefficients

iut + a t uxx + b t |u|2u = 0,

(1)

where a( + 1) = a( ), a + b = 2, a = 2 if [0, 0.5] and a = 0 if [0.5, 1]. Note that at each time-step t = /2 one of the terms is suppressed making this equation equivalent to the split-step approximation of NLS.

Recently Shochet [6] and Grenier [4] developed an averaging technique for the Navier? Stokes equation and Grenier applied that approach to DMNLS to obtain the first-order validity of averaging [9]. Here, we apply his approach to provide a new simple proof of the validity of Fourier split-step method for NLS. With minor modifications this strategy works for other equations of NLS type.

In comparison to the original and more general result by Marsden (see, e.g. [2]), our proof is simpler, and uses more elementary analytic tools: Sobolev spaces and classical averaging method. One difficulty associated with Marsden's approach stems from its generality: in order to apply the main result to NLS one encounters some technical problems in establishing convergence of splitting operators to the vector field of NLS.

Regarding higher order approximation, it is well known since the work of Strang [7] that by using a symmetric split-step scheme, one can improve the convergence to the second order in . Heuristically, one can see it by considering the commutator of the two operators participating in the splitting, as follows. Let ut = (A + B)u be a linear equation with the solution u(t) = e(A+B)t u(0). The exact solution is approximated at t = 1 by

u(1) eA/N eB/N N u(0).

Averaging for split-step scheme

1361

This approximation works because for one step

e(A+B)/N

- eA/N

eB/N

=

1 N2

[A,

B

]

+

O(N

-3),

(2)

where [A, B] = AB -BA, and the error for a finite time interval is of order N ?(1/N 2) N -1.

It was Strang's observation that for the symmetrized split-step

eA/2N eB/N eA/2N N ,

the approximation error for one step in the corresponding expression analogous to (2) is of order N -3 resulting in the error of order N ? (1/N 3) N -2.

Here, we will obtain precise estimates on the convergence of split-step approximations to

the true solutions using an averaging technique. We will also see the difference between the

symmetric and regular splitting in that context.

2. Split-step method for NLS

Decomposing a and b from (1) into the constant and variable parts, we obtain

iut + uxx + |u|2u + p t uxx + q t |u|2u = 0,

(3)

where p = a - 1 and q = b - 1. Note that both p and q have zero mean and p + q = -1. Formally averaging the above equation, we obtain NLS

iut + uxx + |u|2u = 0.

It is well known that the averaged equation NLS has a global solution in H m(R) (for m 0) on an arbitrary interval t [0, T ], see [1]. We then prove the following theorem.

Theorem 1. Let v0(x, t) L([0, T ], H s(R)) (s 4) be a solution of the averaged equation and let u(x, t) be a solution of the full equation with u(x, 0) - v0(x, 0) Hs-4(R) , then there exists 0 > 0 such that for any [0, 0], we have u(x, t) L([0, T ], H s-4(R)) and

sup u(t ) - v0(t ) H s-4(R) C .

t[0,T ]

If the splitting is symmetric and u(x, 0) - v0(x, 0) Hs-6(R) C 2 then quadratic accuracy holds at the integer points

sup u(t ) - v0(t ) H s-6(R) C 2.

t ?N ,t T

Remark 1. The statement of the theorem suggests that in order to achieve higher order accuracy, the initial data must possess higher smoothness.

Proof. Let v0 denote a solution of NLS, which is in H s(R) with s 4. It `approximately' solves the full equation it v0 + xx v0 + |v0|2v0 + p t xx v0 + q t |v0|2v0 = p t xx v0 + q t |v0|2v0,

as the right-hand side of the equation is composed of the rapidly oscillating mean-zero terms. Let us introduce a correction, which would kill the O(1) part of the `error' term on the

right-hand side w1 = i P t xx v0 + i Q t |v0|2v0,

1362

V Zharnitsky

where P , Q are mean-zero anti-derivatives of p, q (note that w1 H s-2(R) and w1 H s-2 C ).

Now, introduce v1 = v0 + w1 (v1 L([0, T ], H s-2)), which is a better approximation of the solution of the full equation

it v1 + v1xx + |v1|2v1 + pv1xx + q|v1|2v1 = R1[v0, t ],

where

R1[v0, t] = (1 + p)xx w1 + (1 + q)(|v1|2v1 - |v0|2v0) - P txx v0 - Q(2|v0|2t v0 + v02t v?0) = (1 + p)xx w1 + (1 + q)(2|v0|2w1 + v02w? 1 + v?0w12 + 2v0|w1|2 + |w1|2w1) - P txx v0 - Q(2|v0|2t v0 + v02t v?0) = i (1 + p)P xxxx v0 + i (1 + p)Q(|v0|2v0)xx + i (1 + q)P 2|v0|2xx v0

+ i 2(1 + q)Q|v0|2v0 - i (1 + q)P v02xxv0 - i (1 + q)Qv03|v0|2 - P txx v0 - Q(2|v0|2t v0 + v02t v?0) + 2F (v0, t/ , ).

Indeed, using that v L([0, T ], H s) and that v0 is a solution of NLS along with the well-

known property of H s in one dimension: f ? g Hs

f

Hs ?

g H s if s >

1 2

for

the

right-hand side we have an estimate2

sup R1 H s-4(R) C

t[0,T ]

if s 4.

Also, one should note that all -order terms in the remainder are mean-zero (use p + q = 1 and P = p, Q = q) on the t/ -scale suggesting that they can be removed at the next step. First, however, we will establish an intermediate result that the split-step scheme provides an

approximate solution. Introducing a mismatch variable = u - v1, we subtract the full equation for u from the

last equation

it + (1 + p)xx + (1 + q)(|v1|2v1 - |u|2u) = R1[v0, t],

or equivalently we have

it + (1 + p)xx + (1 + q)(f (v1, ) + g(v1, )? ) = R1[v0, t],

where f, g are second order polynomials in two variables. We estimate the growth of using

the standard energy estimate. Differentiating the last relation 1, 2, . . . , n times, multiplying by xn? , integrating, and summing the inequalities we obtain

t

2 Hn

C1(

Hn,

v1

Hn)

2 Hn

+

2

|xn ||xnR1| dx

C1(

Hn,

v1

H n ))

2 Hn

+

C2(

H n ) R1

Hn,

where we can take 0 n s - 4. The coefficients C1 and C2 depend on the norms of and v1 and they are bounded if Hn , v1 Hn are bounded. We know that v1 Hn is bounded by the hypothesis and by the properties of the solutions of NLS (v0(t, x) L([0, T ], H s(R)). Assume now that Hn 1, then C1, C2 are bounded. Using this, the bound R1(t) Hs-4 C on t [0, T ] and (0) Hs-4 , we write the corresponding differential inequality

z C1z + C2 ? C ,

where z(t) = (t) Hn . Solving the inequality on the interval [0, t T ] , we obtain z(t) eC1?T (C2 ? C ? T + 1) .

2 In this paper, we use C to denote different constants whenever it does not lead to confusion.

Averaging for split-step scheme

1363

Choosing sufficiently small 0 > 0, we can assume that z(t) 1 on any [0, t we also have that z(t) C , where C depends on T , but is independent of .

Thus, we have

T ] and then

sup (t ) H s-4(R) C ,

t[0,T ]

which implies by a triangular inequality

u(t ) - v0(t ) H s-4 u(t ) - v1(t ) + H s-4 v1(t ) - v0(t ) H s-4

the desired estimate

sup u(t ) - v0(t ) H s-4(R) C ,

t[0,T ]

thus, proving the first part of the theorem. Now, we are going to introduce the next order correction

v2 = v1 + w2

to remove terms of order 2, where w2 is constructed in the same way as before: it is an anti-derivative of the `error term' R1[v0, t, t/ ] with respect to the fast timescale. As we have observed before, the fast timescale dependent coefficients are mean-zero and therefore w2 is periodic in t/ .

It is easy to see that w2 is as smooth as R1 and that it satisfies the estimate

sup w2 H s-4(R)

t[0,T ]

C2

if s 4.

After introducing the correction v2 = v1 + w2, we obtain an approximation of the full equation (3) with 2 mismatch.

it v2 + v2xx + |v2|2v2 + pv2xx + q|v2|2v2 = R2[v0, t ],

where R2 H s-6 and R2 Hs-6 C 2. Proceeding as before, we let = u - v2, and subtracting the equations, we obtain

it + (1 + p)xx + (1 + q)(f (v2, ) + g(v2, )? ) = R2[v0, t].

Repeating the energy estimates, we obtain that (t) Hs-6 C 2 if (0) Hs-6 C 2. For the final estimate we need to use a triangular inequality:

u(t ) - v0(t ) H s-6 u(t ) - v2(t ) + H s-6 v1(t ) - v2(t ) + H s-6 v1(t ) - v0(t ) , H s-6

where the first two terms on the right-hand side (, w2) are already estimated to be of order 2 and the last term w1 is given by

v1(t) - v0(t) = w1 = i P t xxv0 + i Q t |v0|2v0.

It is of order rather than 2 as required. However it is possible to make this term vanish at the integer points (at the beginning and at the end of the splitting `period'). Note, that with the most primitive scheme which we specified on the first page, this term will not vanish (see figure 1(a)). Indeed, the mean-zero anti-derivative of p:

t

1

p-1(t) = p(s) ds -

p(s) ds d

0

00

does not vanish as

1

p-1(0) = -

p(s) ds d = 0.

00

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download